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arXiv:1105.1322 · Physical Review B

Extended quantum U(1) -liquid phase in a three-dimensional quantum dimer model

Olga Sikora, Nic Shannon, Frank Pollmann, Karlo Penc, Peter Fulde

Published Sep 21, 2011• 53 citations
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Abstract

Recently, quantum dimer models have attracted a great deal of interest as a paradigm for the study of exotic quantum phases. Much of this excitement has centered on the claim that a certain class of quantum dimer model can support a quantum $U(1)$-liquid phase with deconfined fractional excitations in three dimensions. These fractional monomer excitations are quantum analogs of the magnetic monopoles found in spin ice. In this paper, we use extensive quantum Monte Carlo simulations to establish the ground-state phase diagram of the quantum dimer model on the three-dimensional diamond lattice as a function of the ratio $\ensuremath{\mu}$ of the potential to kinetic-energy terms in the Hamiltonian. We find that, for ${\ensuremath{\mu}}_{c}=0.75\ifmmode\pm\else\textpm\fi{}0.02$, the model undergoes a first-order quantum phase transition from an ordered ``$\mathsf{R}$ state'' into an extended quantum $U(1)$-liquid phase, which terminates in a quantum critical Rokhsar-Kivelson (RK) point for $\ensuremath{\mu}=1$. This confirms the published field-theoretical scenario. We present detailed evidence for the existence of the $U(1)$-liquid phase and indirect evidence for the existence of its photon and monopole excitations. Simulations are benchmarked against a variety of exact and perturbative results, and a comparison is made of different variational wave functions. We also explore the ergodicity of the quantum dimer model on a diamond lattice within a given flux sector, identifying a new conserved quantity related to transition graphs of dimer configurations. These results complete and extend the previous analysis of O. Sikora et al. [Phys. Rev. Lett. 103, 247001 (2009)].

DOI
10.1103/physrevb.84.115129

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Recently, quantum dimer models have attracted a great deal of interest as a paradigm for the study of exotic quantum phases. Much of this excitement has centered on the claim that a certain class of quantum dimer model can support a quantum U(1)U(1)U(1)-liquid phase with deconfined fractional excitations in three dimensions. These fractional monomer excitations are quantum analogs of the magnetic monopoles found in spin ice. In this paper, we use extensive quantum Monte Carlo simulations to establish the ground-state phase diagram of the quantum dimer model on the three-dimensional diamond lattice as a function of the ratio \ensuremathμ\ensuremath{\mu}\ensuremathμ of the potential to kinetic-energy terms in the Hamiltonian. We find that, for \ensuremathμc=0.75\ifmmode±\else\textpm\fi0.02{\ensuremath{\mu}}_{c}=0.75\ifmmode\pm\else\textpm\fi{}0.02\ensuremathμc​=0.75\ifmmode±\else\textpm\fi0.02, the model undergoes a first-order quantum phase transition from an ordered ``R\mathsf{R}R state'' into an extended quantum U(1)U(1)U(1)-liquid phase, which terminates in a quantum critical Rokhsar-Kivelson (RK) point for \ensuremathμ=1\ensuremath{\mu}=1\ensuremathμ=1. This confirms the published field-theoretical scenario. We present detailed evidence for the existence of the U(1)U(1)U(1)-liquid phase and indirect evidence for the existence of its photon and monopole excitations. Simulations are benchmarked against a variety of exact and perturbative results, and a comparison is made of different variational wave functions. We also explore the ergodicity of the quantum dimer model on a diamond lattice within a given flux sector, identifying a new conserved quantity related to transition graphs of dimer configurations. These results complete and extend the previous analysis of O. Sikora et al. [Phys. Rev. Lett. 103, 247001 (2009)].
We introduce a quantum dimer model on the hexagonal lattice that, in addition to the standard three-dimer kinetic and potential terms, includes a competing potential part counting dimer-free hexagons. The zero-temperature phase diagram is studied by means of quantum Monte Carlo simulations, supplemented by variational arguments. It reveals some new crystalline phases and a cascade of transitions with rapidly changing flux (tilt in the height language). We analyze perturbatively the vicinity of the Rokhsar-Kivelson point, showing that this model has the microscopic ingredients needed for the "devil's staircase" scenario [Eduardo Fradkin et al. Phys. Rev. B 69, 224415 (2004)], and is therefore expected to produce fractal variations of the ground-state flux.
We show that there are two types of resonating-valence-bond (RVB) liquid phases present in three-dimensional quantum dimer models, corresponding to the deconfining phases of U(1)U(1)U(1) and Z2{Z}_{2}Z2​ gauge theories in d=3+1.d=3+1.d=3+1. The former is found on the bipartite cubic lattice and is the generalization of the critical point in the square lattice quantum dimer model found originally by Rokhsar and Kivelson. The latter exists on the nonbipartite face-centered cubic lattice and generalizes the RVB phase found earlier by us on the triangular lattice. We discuss the excitation spectrum and the nature of the ordering in both cases. Both phases exhibit gapped spinons. In the U(1)U(1)U(1) case we find a collective, linearly dispersing, transverse excitation, which is the photon of the low-energy Maxwell Lagrangian and we identify the ordering as quantum order in Wen's sense. In the Z2{Z}_{2}Z2​ case all collective excitations are gapped and, as in d=2,d=2,d=2, the low-energy description of this topologically ordered state is the purely topological BF action. As a byproduct of this analysis, we unearth a further gapless excitation, the pi0n, in the square lattice quantum dimer model at its critical point.
Excitations which carry "fractional" quantum numbers are known to exist in one dimension in polyacetylene, and in two dimensions, in the fractional quantum Hall effect. Fractional excitations have also been invoked to explain the breakdown of the conventional theory of metals in a wide range of three-dimensional materials. However, the existence of fractional excitations in three dimensions remains highly controversial. In this Letter we report direct numerical evidence for the existence of an extended quantum liquid phase supporting fractional excitations in a concrete, three-dimensional microscopic model-the quantum dimer model on a diamond lattice. We demonstrate explicitly that the energy cost of separating fractional monomer excitations vanishes in this liquid phase, and that its energy spectrum matches that of the Coulomb phase in (3+1)-dimensional quantum electrodynamics.
We introduce the quantum dimer-pentamer model (QDPM) on the square lattice. This model is a generalization of the square-lattice quantum dimer model as its configuration space comprises fully packed hard-core dimer coverings as well as dimer configurations containing pentamers, where four dimers touch a vertex. Thus in the QDPM, the fully packed, hard-core constraint of the quantum dimer model is relaxed such that the local dimer number at each vertex is fixed modulo 3, resulting in an exact local Z3{Z}_{3}Z3​ gauge symmetry. We construct a local Hamiltonian for which the Rokhsar-Kivelson (RK) equal superposition state is the exact ground state and has a ninefold topological degeneracy on the torus. Using Monte Carlo calculations, we find no spontaneous symmetry breaking in the RK wave function and that its dimer-dimer correlation function decays exponentially. By doping the QDPM RK state with a pair of monomers, we demonstrate that Z3{Z}_{3}Z3​ electric charges are deconfined. Additionally, we introduce a Z3{Z}_{3}Z3​ magnetic string operator that we find decays exponentially and shows no signatures of magnetic vortex condensation. These numerical results suggest that the ground state of the QDPM is a dimer liquid with Z3{Z}_{3}Z3​ topological order.
We consider a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We focus on a quantum dimer model on the triangular lattice, with doubly periodic edge weights. For simplicity we consider a 2×12\times12×1 periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled ααα. We analytically show that the model exhibits a continuous quantum phase transition at α=3α=3α=3, changing from a topological Z2\mathbb{Z}_2Z2​ quantum spin liquid (α<3α<3α<3) to a columnar ordered state (α>3α>3α>3). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length ξ∝1/∣α−3∣ξ\propto1/|α-3|ξ∝1/∣α−3∣ and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase, which we explain in terms of loop statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class. Additionally, we analytically show that the topological Rényi entropy of order ∞\infty∞ (topological min-entropy) changes from log⁡2\log2log2 for the quantum spin liquid phase α<3α<3α<3, to 000 for the ordered phase α>3α>3α>3, thereby analytically confirming the topological nature of the phase transition.
Two-dimensional Rokhsar-Kivelson (RK) dimer models on bipartite lattices are generally limited to translation-symmetry-broken dimer crystals. We introduce a tensor-product regularization of the dimer Hilbert space that yields a qubit Hamiltonian interpolating from the RK model to the π -flux toric code, thereby accessing a deconfined Z 2 topological liquid. In this framework, the Z 2 liquid descends from a multicritical U ( 1 ) spin liquid through condensation of a charge-2 Higgs field, thus avoiding confinement. Using iDMRG together with low-energy field theory, we determine a phase diagram containing two continuous quantum phase transitions—a 3 D XY * transition between the Z 2 liquid and the columnar/plaquette-VBS, and a quantum Lifshitz transition between two dimer crystals—alongside a first-order transition between the staggered crystal and the Z 2 liquid. Our field theory suggests a deconfined multicritical point described by an Abelian Higgs model with dynamical critical exponent, z = 2 , where the three transitions meet, highlighting the interplay of fractionalization and emergent gauge fluctuations.